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Theorem rngsubdi 15635
Description: Ring multiplication distributes over subtraction. (subdi 9399 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
Hypotheses
Ref Expression
rngsubdi.b  |-  B  =  ( Base `  R
)
rngsubdi.t  |-  .x.  =  ( .r `  R )
rngsubdi.m  |-  .-  =  ( -g `  R )
rngsubdi.r  |-  ( ph  ->  R  e.  Ring )
rngsubdi.x  |-  ( ph  ->  X  e.  B )
rngsubdi.y  |-  ( ph  ->  Y  e.  B )
rngsubdi.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
rngsubdi  |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( ( X 
.x.  Y )  .-  ( X  .x.  Z ) ) )

Proof of Theorem rngsubdi
StepHypRef Expression
1 rngsubdi.r . . . 4  |-  ( ph  ->  R  e.  Ring )
2 rngsubdi.x . . . 4  |-  ( ph  ->  X  e.  B )
3 rngsubdi.y . . . 4  |-  ( ph  ->  Y  e.  B )
4 rnggrp 15596 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
51, 4syl 16 . . . . 5  |-  ( ph  ->  R  e.  Grp )
6 rngsubdi.z . . . . 5  |-  ( ph  ->  Z  e.  B )
7 rngsubdi.b . . . . . 6  |-  B  =  ( Base `  R
)
8 eqid 2387 . . . . . 6  |-  ( inv g `  R )  =  ( inv g `  R )
97, 8grpinvcl 14777 . . . . 5  |-  ( ( R  e.  Grp  /\  Z  e.  B )  ->  ( ( inv g `  R ) `  Z
)  e.  B )
105, 6, 9syl2anc 643 . . . 4  |-  ( ph  ->  ( ( inv g `  R ) `  Z
)  e.  B )
11 eqid 2387 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
12 rngsubdi.t . . . . 5  |-  .x.  =  ( .r `  R )
137, 11, 12rngdi 15609 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  ( ( inv g `  R ) `  Z
)  e.  B ) )  ->  ( X  .x.  ( Y ( +g  `  R ) ( ( inv g `  R
) `  Z )
) )  =  ( ( X  .x.  Y
) ( +g  `  R
) ( X  .x.  ( ( inv g `  R ) `  Z
) ) ) )
141, 2, 3, 10, 13syl13anc 1186 . . 3  |-  ( ph  ->  ( X  .x.  ( Y ( +g  `  R
) ( ( inv g `  R ) `
 Z ) ) )  =  ( ( X  .x.  Y ) ( +g  `  R
) ( X  .x.  ( ( inv g `  R ) `  Z
) ) ) )
157, 12, 8, 1, 2, 6rngmneg2 15633 . . . 4  |-  ( ph  ->  ( X  .x.  (
( inv g `  R ) `  Z
) )  =  ( ( inv g `  R ) `  ( X  .x.  Z ) ) )
1615oveq2d 6036 . . 3  |-  ( ph  ->  ( ( X  .x.  Y ) ( +g  `  R ) ( X 
.x.  ( ( inv g `  R ) `
 Z ) ) )  =  ( ( X  .x.  Y ) ( +g  `  R
) ( ( inv g `  R ) `
 ( X  .x.  Z ) ) ) )
1714, 16eqtrd 2419 . 2  |-  ( ph  ->  ( X  .x.  ( Y ( +g  `  R
) ( ( inv g `  R ) `
 Z ) ) )  =  ( ( X  .x.  Y ) ( +g  `  R
) ( ( inv g `  R ) `
 ( X  .x.  Z ) ) ) )
18 rngsubdi.m . . . . 5  |-  .-  =  ( -g `  R )
197, 11, 8, 18grpsubval 14775 . . . 4  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  R
) ( ( inv g `  R ) `
 Z ) ) )
203, 6, 19syl2anc 643 . . 3  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  R
) ( ( inv g `  R ) `
 Z ) ) )
2120oveq2d 6036 . 2  |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( X  .x.  ( Y ( +g  `  R
) ( ( inv g `  R ) `
 Z ) ) ) )
227, 12rngcl 15604 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
231, 2, 3, 22syl3anc 1184 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  e.  B )
247, 12rngcl 15604 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
251, 2, 6, 24syl3anc 1184 . . 3  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
267, 11, 8, 18grpsubval 14775 . . 3  |-  ( ( ( X  .x.  Y
)  e.  B  /\  ( X  .x.  Z )  e.  B )  -> 
( ( X  .x.  Y )  .-  ( X  .x.  Z ) )  =  ( ( X 
.x.  Y ) ( +g  `  R ) ( ( inv g `  R ) `  ( X  .x.  Z ) ) ) )
2723, 25, 26syl2anc 643 . 2  |-  ( ph  ->  ( ( X  .x.  Y )  .-  ( X  .x.  Z ) )  =  ( ( X 
.x.  Y ) ( +g  `  R ) ( ( inv g `  R ) `  ( X  .x.  Z ) ) ) )
2817, 21, 273eqtr4d 2429 1  |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( ( X 
.x.  Y )  .-  ( X  .x.  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   .rcmulr 13457   Grpcgrp 14612   inv gcminusg 14613   -gcsg 14615   Ringcrg 15587
This theorem is referenced by:  2idlcpbl  16232  nrgdsdi  18572  nrginvrcnlem  18597  ply1divmo  19925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-plusg 13469  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mgp 15576  df-rng 15590  df-ur 15592
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